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Inmathematics, the support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. If the domain of
is a topological space, then the support of
is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.
Suppose that is a real-valued function whose domain is an arbitrary set
The set-theoretic supportof
written
is the set of points in
where
is non-zero:
The support of is the smallest subset of
with the property that
is zero on the subset's complement. If
for all but a finite number of points
then
is said to have finite support.
If the set has an additional structure (for example, a topology), then the support of
is defined in an analogous way as the smallest subset of
of an appropriate type such that
vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than
and to other objects, such as measuresordistributions.
The most common situation occurs when is a topological space (such as the real lineor
-dimensional Euclidean space) and
is a continuous real- (orcomplex-) valued function. In this case, the supportof
,
, or the closed support of
, is defined topologically as the closure (taken in
) of the subset of
where
is non-zero[1][2][3] that is,
Since the intersection of closed sets is closed, is the intersection of all closed sets that contain the set-theoretic support of
For example, if is the function defined by
then
, the support of
, or the closed support of
, is the closed interval
since
is non-zero on the open interval
and the closure of this set is
The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that (or
) be continuous.[4]
Functions with compact support on a topological space are those whose closed support is a compact subset of
If
is the real line, or
-dimensional Euclidean space, then a function has compact support if and only if it has bounded support, since a subset of
is compact if and only if it is closed and bounded.
For example, the function defined above is a continuous function with compact support
If
is a smooth function then because
is identically
on the open subset
all of
's partial derivatives of all orders are also identically
on
The condition of compact support is stronger than the condition of vanishing at infinity. For example, the function defined by
vanishes at infinity, since
as
but its support
is not compact.
Real-valued compactly supported smooth functions on a Euclidean space are called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.
Ingood cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of limits, for any any function
on the real line
that vanishes at infinity can be approximated by choosing an appropriate compact subset
of
such that
for all
where
is the indicator functionof
Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.
If is a topological measure space with a Borel measure
(such as
or a Lebesgue measurable subset of
equipped with Lebesgue measure), then one typically identifies functions that are equal
-almost everywhere. In that case, the essential support of a measurable function
written
is defined to be the smallest closed subset
of
such that
-almost everywhere outside
Equivalently,
is the complement of the largest open set on which
-almost everywhere[5]
The essential support of a function depends on the measure
as well as on
and it may be strictly smaller than the closed support. For example, if
is the Dirichlet function that is
on irrational numbers and
on rational numbers, and
is equipped with Lebesgue measure, then the support of
is the entire interval
but the essential support of
is empty, since
is equal almost everywhere to the zero function.
In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so is often written simply as
and referred to as the support.[5][6]
If is an arbitrary set containing zero, the concept of support is immediately generalizable to functions
Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family
of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily
is the countable set of all integer sequences that have only finitely many nonzero entries.
Functions of finite support are used in defining algebraic structures such as group rings and free abelian groups.[7]
Inprobability theory, the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra, rather than on a topological space.
More formally, if is a random variable on
then the support of
is the smallest closed set
such that
In practice however, the support of a discrete random variable is often defined as the set
and the support of a continuous random variable
is defined as the set
where
is a probability density functionof
(the set-theoretic support).[8]
Note that the word support can refer to the logarithm of the likelihood of a probability density function.[9]
It is possible also to talk about the support of a distribution, such as the Dirac delta function on the real line. In that example, we can consider test functions
which are smooth functions with support not including the point
Since
(the distribution
applied as linear functionalto
) is
for such functions, we can say that the support of
is
only. Since measures (including probability measures) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way.
Suppose that is a distribution, and that
is an open set in Euclidean space such that, for all test functions
such that the support of
is contained in
Then
is said to vanish on
Now, if
vanishes on an arbitrary family
of open sets, then for any test function
supported in
a simple argument based on the compactness of the support of
and a partition of unity shows that
as well. Hence we can define the supportof
as the complement of the largest open set on which
vanishes. For example, the support of the Dirac delta is
InFourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a smooth function.
For example, the Fourier transform of the Heaviside step function can, up to constant factors, be considered to be (a function) exceptat
While
is clearly a special point, it is more precise to say that the transform of the distribution has singular support
: it cannot accurately be expressed as a function in relation to test functions with support including
Itcan be expressed as an application of a Cauchy principal value improper integral.
For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).
An abstract notion of family of supports on a topological space suitable for sheaf theory, was defined by Henri Cartan. In extending Poincaré dualitytomanifolds that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example Alexander–Spanier cohomology.
Bredon, Sheaf Theory (2nd edition, 1997) gives these definitions. A family of closed subsets of
is a family of supports, if it is down-closed and closed under finite union. Its extent is the union over
Aparacompactifying family of supports that satisfies further that any
in
is, with the subspace topology, a paracompact space; and has some
in
which is a neighbourhood. If
is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.